The study of general Dirichlet series has its origins in the works  -  of E. Cahen, J. Hadamard, E. Landau, H. Bohr, G. H. Hardi, M. Riesz, T. Kojima, M. Kuniyeda, G. Valiron, etc. A lot of contemporary mathematicians created a diversified theory of general Dirichlet series, some insisting on the connection with the Laplace-Stieltjes transforms  -  as J. Yu, Y. Kong, S. Daochun, X. Luo, Y. Yan, C. Singhal, G. S. Srivastava, etc., others as P.K. Kamthan, S. K. Shing Gautam, L.H. Khoi,   endowing them with some topological structures, extending to them the Nevanlinna theory (too many to be cited) or dealing with vector valued Dirichlet series  -  as A. Defant, M. Maestre, D. Perez-Garcia, J. Bonet, B.L. Srivastava, A. Sharma, G. S. Srivastava, etc.
We will use normalized series defined as follows. To any sequence of complex numbers and any increasing sequence such that we associate the series
is called the type of the series (1) and the series defined by the same will be called series of the same type. When , we obtain the ordinary (proprement dites,  ) Dirichlet series.
Suppose that and are such that the abscissa of convergence (see   ) of the series (1)
is finite. (2)
Then (see  ) is an analytic function in the half plane , where is the abscissa of uniform convergence of (1), and where is at most with
For the ordinary Dirichlet series and it is known that in the case of Dirichlet L-series defined by imprimitive Dirichlet characters and , while in the case of primitive characters .
In general, when then , and the series (1) is an analytic function in the half plane .
Suppose that this function can be continued analytically to the whole complex plane, except possible at which is a simple pole. We keep the notation for the function obtained by analytic continuation.
With the exception of a discrete set of points from the complex plane, this function is locally injective, i.e. it maps conformally and hence bijectively small neighbourhoods of every point onto some domains. Enlarging these neigh- bourhoods, the image domains get bigger. How big can they get? The answer is: they become the whole complex plane with some slits (see     ). A region with this property is called fundamental domain of .
Our aim is to show that the complex plane can be divided into a countable number of sets whose interiors are fundamental domains of this function. We have done this previously for the Riemann Zeta function   as well as for Dirichlet L-functions  . To do the same thing for functions obtained by analytic continuations of general Dirichlet series we made  the assumption that (1) satisfies a Riemann type of functional equation. We will show next that such a strong assumption is not necessary and we can obtain the same result by using just elementary properties of the conformal mapping.
It has been proved  that for normalized series (1) the limit is uniform with respect to . This means that for any there is such that implies , therefore the whole half plane is mapped by into a disc centred at of radius .
An immediate consequence of this fact is that the abscissa of convergence of a normalized series is less than . Moreover, for there is no zero of in the half plane . Also, if satisfies a Riemann type of functional equation, then there cannot be non trivial zeros of in the half plane neither.
2. Pre-Images of Lines and Circles
Suppose that, for a point with the function has a real value . The continuation from along the interval , is a curve such that when tends to 1, we have that tends to on , or there is a point such that and the con- tinuation can be carried along the whole real axis giving rise to a curve . We will show later that when tends to on the real axis then tends to on , or on , in other words these curves cannot remain in a right half plane.
Let us notice first that are transcendental functions and is an essential singular point for them. The value cannot be a lacunary
value for since then would be a non polynomial integer
function for which is an essential singular point and hence would be also an essential singular point for , which is not true. Then, by the Big Picard Theorem, has infinitely many zeros in every neigh- bourhood of .
Given a bounded region of the plane, there is such that the pre-image of the circle centred at the origin and of radius has only disjoint com- ponents, which are closed curves containing each one a unique zero of belonging to that region. When increases those curves expand and they can touch one another at some points (see Figure 1(b)). These are branch points of the function, since in every neighbourhood of the function takes at least twice any value on the image circle. Therefore the derivative of cancels at . It is obvious that any zero of the derivative, which is not a zero of the function itself, can be obtained in this way. Indeed, if is such a zero, we can take and necessarily at least two components of the pre-image of will pass through .
What happens with those components of the pre-image of a circle when ? We have proved  , Theorem 1 that there is at least one unbounded component of the pre-image of the unit circle. That proof did not use the as- sumption of satisfying a Riemann type of functional equation and therefore it is true for any function .
Let us notice that two curves and cannot intersect each other. Indeed, if would be a common point of these curves, then when moves on the interval between and 1 the point describes an unbounded curve which bounds a domain mapped by onto the complex plane with a slit alongside the interval . That domain should contain
Figure 1. Pre-images of circles and the birth of a strip.
a pole of which is not true. Therefore an intersection point of the two curves cannot exist and consecutive curves and bound infinite strips . We suppose that is the strip containing the point and for every integer , the curve is situated above . Figure 1(e) illustrates the birth of a strip when the pre-image of a ray making a small angle with the positive real half axis is taken and then we let .
We have also proved  , Theorem 2 that every unbounded component of the pre-image by of the unit circle is contained between two con- secutive curves and and vice-versa, if between two con- secutive curves and there is a unique unbounded component of the pre-image of the unit circle. It has been shown that the respective component does not intersect any one of these curves. For every strip contains also a unique curve which is mapped bijectively by onto the interval of the real axis, as well as a certain number of curves , which are mapped bijectively by onto the whole real axis. There are infinitely many strips covering the whole complex plane  , Theorem 4. Therefore, there are infinitely many unbounded components of the pre-image by of the unit circle. Some strips can contain also bounded components of the pre-image of the unit circle as well as bounded components of the pre-image of with . Figure 1(d) portrays the strip of containing two components of the pre-image of the unit circle: the unbounded one containing two zeros and the other bounded, containing one zero.
The use of the pre-image of the real axis can be traced back to Speiser’s work  on the Riemann Zeta function. After that the pre-image of the real axis does not appear any more in literature as a tool except for the paper of Arias-de-Reina  , who revisits Speiser’s theorem. In the same year John Derbyshire uses both: the pre-image of the real axis and that of the imaginary axis in his popular book Prime Obsession and declares that they are at the heart of that book. The classification of the components of the pre-image of the real axis by the Riemann Zeta function appears for the first time in  , where the strips are also introduced and a method is devised of partitioning them into fundamental domains. Later, such a classification has been extended to Dirichlet L-functions and finally to functions defined by general Dirichlet series.
A different approach, namely that of phase diagram, has been used by Elias Wegert for visual exploration of complex functions   . Applied to the Riemann Zeta function, his phase plots revealed interesting patterns pertaining to the universality property of that function. It is known that such a property extends to more general Dirichlet series and probably it can complement our fundamental domains approach.
An even better way to visualize the complexity of conformal mappings by analytic functions of a complex variable is to use an orthogonal mesh in the z-plane formed with rays issuing from the origin and circles centred at the origin. Moreover, a spectre of colors can be superposed to the mesh as seen in Figures 2(a)-(c). By taking the pre-image of that mesh we obtain a coloured orthogonal mesh in the s-plane in which the color of every point coincides with that of its
Figure 2. Colour-visualization of the conformal mapping by .
image. In this way we can locate the corresponding points in the two planes and have also a global view of the mapping. The pre-image of the real axis and the -strips are still identifiable. Figures 2(d)-(f) above illustrates the mapping by the Riemann Zeta function in the rectangles , and . By comparing them, one can notice the increasing number of zeros in the strips with increasing . The pattern we can see here is proper to any function .
Theorem 1 No zero of or of can belong to a curve .
Proof. The affirmation of the theorem is obvious for the zeros of since 0 does not belong to the interval of the real axis. A more intricate argument guarantees that the same is true for the zeros of the derivative of . Indeed, even if , no bounded component of the pre-image of can reach , despite of the fact that intersects the interval . Indeed, in the contrary case, the respective component should intersect at lest twice, or it should be tangent to it, fact which requires that intersects the interval the same number of times or to be tangent to it, which is not possible. The fact that intersects the interval has as effect the pre-image of intersecting the curves (and not ). It results that no two bounded components of the pre-image of can meet on , neither can one of these components meet on an unbounded component of the pre-image of into a zero of .
Remark: Theorem 1 does not imply that cannot have zeros on some . Such zeros appeared as possible for the Dirichlet L-function as seen in Figure 3 when has approximately the values 169.2 and 179.2. However, we suspected that this was due to the poor resolution of the picture and indeed, when we zoomed on the respective points, we obtained configu- rations which show clearly that does not cancel there. However, as we will see later, the possibility of such zeros cannot be excluded.
We have seen that the unbounded components of the pre-image of the unit circle do not intersect any . On the other hand the bounded components of the pre-image of the unit circle intersect curves at points where
. In the same way the bounded components of the pre-image of
with will intersect at points where . However the story of the unbounded components of the pre-image of is a little more complicated.
When increasing past 1 all the unbounded components of the pre-image of fuse together into a unique unbounded curve intersecting every curve , hence they do not generate by this fusion zeros of . Indeed, since the mapping of onto the interval is bijective, there should be a unique point on every such that . The continuation over from each one of these points can be made clockwise and counter clockwise into , respectively , for every , giving rise to that unbounded curve. The final conclusion is that does not cancel on any . ■
Figure 3. Possible zeros of on which are not double zeros of .
Given any bounded domain in the plane , we can take close enough to 1 such that does not touch any bounded component of the pre-image of included in that domain. However, for bigger values of the curve comes into contact with bounded components which were turning around one or several zeros of , fusing with them and getting to the left of those zeros, hence intersecting also the curves which contain the respective zeros. The curve is orthogonal to every component of the pre- image of the real axis if it intersects that component at a point where does not cancel.
Since a point turning around the origin in the same direction on an arbitrary circle centred at the origin will meet consecutively the positive and the negative real half axis, the components of the pre-image of (including when ) should meet consecutively the pre-image of the positive and the negative half axis (coloured differently). This is   the so-called color alternating rule.
An immediate consequence of this rule is that in every strip the first and the last curve should be such that the pre-image of the negative real half axis faces the corresponding . Then, consecutive have the same orien- tation, as long as they are on the same side of . A zero of being on and not being a zero of (where the colors are changing) implies that different components of the pre-image of a circle fuse (for a value of ) on . The respective components must obey the color alternating rule for every and we realize that after fusion the component can be such that the rule is still in force. Indeed, if and have the same color at then this curve can continue to meet the respective color without affecting the color alternation by simply switching the branches of and which come to . However, as we will see next, the position of with respect to the two zeros of on these curves cannot be arbitrary.
Theorem 2 For different of 0, there is no zero of the derivative of at the left of the leftmost zero of in
Proof. We can reproduce the proof of Theorem 1 from  for arbitrary functions . Let be the leftmost zero of from and suppose that the simple zero of is a progenitor of , i.e. a component of the pre-image of the circle with contains the point . It is obviously enough to deal with the case where , i.e. where is above . Then (indicated in Figure 4 as , since it has been partially computer generated by a Dirichlet L-function) belongs to the upper half plane and , where .
The pre-image of the ray determined by contains two curves which are orthogonal at . The angles at are doubled by , hence the four arcs of the pre-image of that ray make angles of , , and with a horizontal line whose image passes through . The angle made by the second arc (which ends in ) with this line is less than the angle made by the tangent to the pre-image of the respective ray at any point between and with the same horizontal line. If , then there must be a point on that arc for which , therefore , which is absurd. ■
Corollary: If satisfies RH, then the zeros of from every strip are at the right side of the critical line. When is the Riemann
Figure 4. The location of the zeros of .
Zeta function this corollary represents the Speiser’s theorem  .
The existence of multiple zeros of functions obtained by analytic con- tinuations of Dirichlet series has been documented (probably for the first time) in  , where double zeros of a linear combination of Dirichlet L-functions have been found (see Figure 5 below).
We have shown that all those double zeros are located on the critical line. In this example the function is and the double zero is obtained for the approximate value of of . The double zeros we have found for all the functions of this type were located at the intersection of and or of and . We can make now a much more general affirmation about the multiple zeros of functions obtained by analytic con- tinuation of general Dirichlet series.
Theorem 3 In every strip of a function this function has at most one double zero. Such a zero is found at the intersection of and or of and . There is no multiple zero of in and hence no zero of a higher order than two of .
We need to postpone the proof of this theorem for a while.
3. Intertwining Curves
When studying functions it is useful to consider besides the planes and also a plane , where . Sometimes the planes and will be identified in order to make more obvious certain relations between the configurations defined by the two functions in the respective planes. The configurations we have in view are pre-images by both and by
Figure 5. A double zero of a linear combination of dirichlet L-functions.
, of some curves or domains.
Regarding the pre-image by and by of the real axis it has been found   that the components of these pre-images are paired in such a way that only the components of the same pair can intersect each other. The respective pairs form the so-called intertwining curves.
Three kinds of intertwining curves have been distinguished     , namely:
1) and , , which are mapped bijectively by and by onto the interval , respectively ,
2) and , , which are mapped bijectively by and by onto the interval , respectively
3) and , , , (a finite set of integers), which are mapped bijectively by , respectively by onto the whole real axis.
Theorem 4 The intertwining curves touch each other at the points where the tangent to , respectively is horizontal. Vice-versa, if at a point of such a curve the tangent is horizontal, then a component of the pre-image of the real axis by passes also through that point.
Proof. Indeed, suppose that is the equation of a curve or of a curve such that . Then (see Figure 6 below)
Figure 6. Intertwining curves of for .
The Equation (4) shows that the argument of is opposite to that of , therefore they cancel simultaneously, and when one is , the other should be . Yet these values of the argument of a point mean that the respective point is on the real axis, therefore belongs to both pre-images of the real axis, by and which completely proves the theorem.
Remark: The Theorem 4 is a corollary of a much more general property which says that if is an analytic function in a domain of the complex plane and is the image by of a smooth curve , then denoting by the pre-image by of , at every point where and intersect each other we have , where .
Indeed, suppose that the planes and are identified, where and write for the curve . Then at an intersection point of and we have and implies , hence , etc.
We can show now that:
Theorem 5 No strip can be included in a right half plane.
Proof. The formula (4) written for tells us that when tends to then must tend to 0. Yet, there is no zero of on any . On the other hand, is an unlimited continuation of alongside , hence when tends to we have that tends to not to 0 and this is a contradiction. The conclusion is that the geometry of the pre-image of the real axis is in the whole complex plane similar to that we can see in a bounded region of the plane in Figure 2, Figure 3, Figure 6, Figure 7. ■
Summarizing these facts and having in view  we can say:
Theorem 6 The variable takes any real value on every curve . Con- secutive curves form infinite strips which are mapped (not necessarily bijectively) onto the whole complex plane with a slit alongside the interval of the real axis. If contains zeros of the function , then it will contain zeros of . For any given , the function either has a finite number of zeros in , or . If satisfies a Riemann type of functional equation, every strip , contains a finite number of zeros of this function. The strip may contain infinitely many zeros of .
Proof. We only need to justify the numbers and to which the theorem makes reference. We have seen that every zero of is obtained when two components of the pre-image of some circle centred at the origin and of radius come into contact. If we consider the zeros of as the leafs of a binary tree whose internal nodes are obtained in this way, that tree is a complete binary tree and it is known that it must have exactly internal nodes. Figure 7 illustrates this situation for ten strips of the Riemann Zeta function. The location of the zeros of and those of its first two derivative
Figure 7. The zeros of and of its first two derivatives for .
are also clearly indicated. If there is a double zero of in , then the respective leaf is counted twice as a leaf and once as an internal node, etc. When the number of zeros in is infinite we must have , otherwise would have an accumulation point in and hence would be identically zero. ■
We can prove now also the Theorem 3.
Proof. Exactly 2 curves and should meet at a double zero of belonging to . But is also a simple zero of and therefore a curve, say should pass through . Yet, intertwines with a curve and unless this curve is , which does not contain any zero of , the point would be a double zero of for , and this is impossible.
Hence , necessarily one of the curves passing through is . This shows that there can be only one double zero in and the other curve passing through in that case is either , or .
We notice that the curve does not contain any zero of , hence it cannot pass through a multiple zero of order of . Since every curve passing through that zero has an intertwining curve defined by the second derivative of , the respective point should be a multiple zero of order of this second derivative, which is absurd. Therefore has no multiple zero and then cannot have any zero of the higher order than 2. ■
4. Fundamental Domains
For , the curves as well as are parabola like curves with branches extending to infinity as . Therefore we can distinguish between the interior and the exterior of such curves. They can be viewed as oriented curves, with the same orientation as the real axis whose components of the pre-image they are. By the same rule and , as well as are also oriented, the positive orientation of and of being from the right to the left, while that of , is from the left to the right. Some curves , can contain in interior some other curves of the same type (embraced curves) and by the color alternating rule the orientation of the embracing curve and that of the embraced curves must be different. We did not find any instance where an embraced curve is in turn embracing, yet there is no reason to believe that such a situation is impossible.
Theorem 7 If the strip contains zeros counted with multiplicities, then can be partitioned into sub-strips which are fundamental do- mains for
Proof. Suppose that curves and are containing the simple zeros and and two components of the pre-image of a circle which are going around each one of these zeros touch at a point . This is a zero of . The pre-image of the segment of line connecting and has as component an arc connecting the points on the two curves where and passing through . If one of these curves is then the respective point is and is an unbounded curve. The strip bounded by this curve and the branches of and corre- sponding to the interval , turns out to be a fundamental domain of . Indeed, is mapped conformally by onto the whole complex plane with a slit alongside the interval followed by a slit alongside the segment from to . If embraces then is bounded to the right.
Suppose now that one of the two zeros is a double zero of . We know that must pass through that zero and the part of corresponding to the interval becomes part of the boundaries of adjacent fundamental domains. One of them will have as image the complex plane with a slit alongside the positive real half axis and for the other one a slit from to should be added, where is 1 or −1.
We know that cannot have a higher order of multiplicity and therefore the cases analysed exhaust all the possibilities. Having in view Theorem 6, we conclude that the strip can be always divided into fundamental domains. ■
Figure 8 below illustrates this theorem for the case of the Riemann Zeta
(a) (b) (c)
Figure 8. Two fundamental domains and their images.
function and the strip .
As seen in the case of Dirichlet L-functions (including the Riemann Zeta function) the strip has infinitely many zeros, yet following the same technique we can divide it into infinitely many fundamental domains.
If four different colors are used, say color 1 and 2 for the pre-images by of the positive and of the negative real half axis and 3 and 4 for the pre-images by of the same half axes, then two simple topological facts can be established  :
a) The color alternating rule, which states that as a point turns indefinitely in the same direction on a circle centred at the origin, the pre-images of this point by each one of the functions and will meet alternatively the colors 1 and 2, respectively 3 and 4.
b) The color matching rule, which states that when intertwining curves meet each other, then if these are not and color 1 will always meet color 4 and color 2 will always meet color 3. Only the curves and can intersect each other at points where color 2 can meet color 4.
The series (1) which are Euler products display special important properties.
5. Euler Products
It is known that the Dirichlet L-functions are meromorphic continuations of ordinary Dirichlet series defined by Dirichlet characters and these series can be expressed as Euler products. This property is a corollary of the fact that the Dirichlet characters are totally multiplicative functions (see for example  . Yet the property of being total multiplicative can be extended to general Dirichlet series, as done in  , and therefore some of the general Dirichlet series (see the details in  ) can also be written as Euler products:
where is the set of prime numbers. This convention will be kept in the following for all the products and sums involving the subscript . The product has the same abscissa of convergence as the series itself.
Looking for counterexamples to the Grand Riemann Hypothesis (GRH), some Dirichlet series satisfying a Riemann type of functional equation have been found, whose analytic continuation exhibit off critical line non trivial zeros, namely the Davenport and Heilbronn type of functions and linear combinations of L-functions satisfying the same functional equation. Although these are not counterexamples to GRH, their study allowed us to draw interesting conclusions. We have seen in  that if does not satisfy the GRH, then for every two distinct non trivial zeros and there is , such that , where , i.e. the derivative of cancels at a point of the interval deter- mined by and . Moreover, .
Let us rephrase and give a simplified proof to  , Theorem 3.
Theorem 8 Suppose that the function (5) satisfies a Riemann type of func- tional equation and the respective series has the abscissa of convergence . Then for every non trivial zero of we have .
Proof. Suppose that there is a zero of for which . Then, due to the functional equation, is also a zero of (see Figure 9 below). There is such that in the components and of the pre-image of the disc centred at the origin and of radius containing the respective zeros the function is injective. Then we can define the function as follows:
The function can be continued as an analytic involution of the union of the fundamental domains and of containing the respective zeros. The boundaries of the domains and have a common component and the union is a simply connected domain . The function can be continued to an analytic involution of having as a fixed point. We have in , in particular and . Moreover, in . Let us define the function by
Since the numerator and the denominator of are analytic functions in and the denominator cancels only at and , the function is analytic in except at these two points. Since , we have that for not equal to one of these points. Yet they are removable singularities, and we can set in . By the formula (5) we have
for sÎH, > σc. In particular,
Figure 9. Symmetric zeros with respect to the critical line.
The arguments of these ratios represent the angles under which the segment between and is seen from the point on the unit circle. If a Ramanujan type condition is fulfilled, namely and are such that for every we have , then the respective angles tend to zero as . This appears to be a necessary condition for the convergence of the series.
However, the condition is implicitly satisfied since we know that is well defined in the domain . The series is a continuous function of , yet it can take only integer multiple values of , which is possible only if it is a constant. Since the series can remain constant only if there
is p0 such that for p > p0 we have
This can happen in two situations: either the three points , and are collinear, or and in this last case , i.e. . In the ﬁrst case, a shift in t will not aﬀect the real part of the zeros, yet it will destroy the collinearity of the three points and therefore this situation can be ignored. The ﬁnal conclusion is that cannot have any non trivial zero with strictly greater than 1/2. The zero can be either a simple zero and then we have only one fundamental domain , or a double zero and then it is the fixed point of the involution . In both cases it is located on the critical line, which completely proves the theorem.
We need to point out the fact that Figure 9 above illustrates a situation in which the Equation (5) is not satisfied.
Remark: In all of our publications we understood by trivial zeros of an L- function those zeros which can be trivially computed. In this respect, the non trivial zeros of the alternating Zeta function are the same as those of and satisfies the conditions of Theorem 8, therefore its non trivial zeros are located on the critical line. Thus, RH is also true for .
Similarly, the non trivial zeros of a Dirichlet L-function defined by the principal character modulo are the non trivial zeros of the Riemann Zeta function.
Also, the non trivial zeros of a Dirichlet L-function induced by an imprimitive character are the non trivial zeros of the function defined by the associated primitive character. Consequently, the RH for any Dirichlet L-function is fulfilled. With this understanding of the concept of non trivial zeros, Theorem 8 represents the proof of GRH for a wide class of functions.
The author is very much thankful to Florin Alan Muscutar for his contribution with computer generated graphics.
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